where \(h (T, \mathcal {C}) = \lim \limits _{n \rightarrow \infty } \frac{1}{n} H \left( \bigvee _{j=0}^{n-1} T^{-j} \mathcal {C} \right)\) .
It is well-known (cf. [1]} or [2]}) that if \(H(\mathcal {C}) < \infty \) , then \(h(T, \mathcal {C}) \le H(\mathcal {C})\)  and if \(\mathcal {A}\)  generates \(\sigma \) -algebra \(\mathcal {B}\) , then \(h(T) = h(T, \mathcal {A})\) . So it follows that any system belonging to \(\mathcal {T}\)  has finite entropy.

